On resonant interactions of gravity-capillary waves without energy exchange

Malik Chabane, Wooyoung Choi

Research output: Contribution to journalArticlepeer-review

7 Scopus citations


We consider resonant triad interactions of gravity-capillary waves and investigate in detail special resonant triads that exchange no energy during their interactions so that the wave amplitudes remain constant in time. After writing the resonance conditions in terms of two parameters (or two angles of wave propagation), we first identify a region in the two-dimensional parameter space, where resonant triads can be always found, and then describe the variations of resonant wavenumbers and wave frequencies over the resonance region. Using the amplitude equations recovered from a Hamiltonian formulation for water waves, it is shown that any resonant triad inside the resonance region can interact without energy exchange if the initial wave amplitudes and relative phase satisfy the two conditions for fixed point solutions of the amplitude equations. Furthermore, it is shown that the symmetric resonant triad exchanging no energy forms a transversely modulated traveling wave field, which can be considered a two-dimensional generalization of Wilton ripples.

Original languageEnglish (US)
Pages (from-to)528-550
Number of pages23
JournalStudies in Applied Mathematics
Issue number4
StatePublished - May 2019

All Science Journal Classification (ASJC) codes

  • Applied Mathematics


  • gravity-capillary waves
  • resonant triad interactions
  • symmetric Wilton ripples


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